Gross-inspired mirror conjecture for cominuscule fixed points

Let $\alpha_{i_1},\dots,\alpha_{i_k}$ be distinct cominuscule roots. Suppose the corresponding sections $b_{i_j}$ of $L_{\alpha_{i_j}}K$ have $m$ distinct zeros $x_1,\dots,x_m$, while $b_i$ has no zero for every other simple root. Let the Langlands dual group be $G^{\vee}$, and let the corresponding minuscule representation be the one associated with the relevant dual weight. Cominuscule mirror conjecture. The fixed point is very stable, and the mirror of its upward flow is the tensor product, over the points $x_j$, of the vector bundles associated with the universal principal $G^{\vee}$-bundle in the corresponding minuscule representation. This is suggested by the identification of the Lefschetz action on the cohomology of a flag variety with the principal $\operatorname{SL}_2$ representation of the Langlands dual group. The claimed very stability and mirror identification are not proved in the supplied text.